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Theorem dfrnf 4748
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfrnf.1 𝑥𝐴
dfrnf.2 𝑦𝐴
Assertion
Ref Expression
dfrnf ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 4695 . 2 ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2 nfcv 2256 . . . . 5 𝑥𝑣
3 dfrnf.1 . . . . 5 𝑥𝐴
4 nfcv 2256 . . . . 5 𝑥𝑤
52, 3, 4nfbr 3942 . . . 4 𝑥 𝑣𝐴𝑤
6 nfv 1491 . . . 4 𝑣 𝑥𝐴𝑤
7 breq1 3900 . . . 4 (𝑣 = 𝑥 → (𝑣𝐴𝑤𝑥𝐴𝑤))
85, 6, 7cbvex 1712 . . 3 (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
98abbii 2231 . 2 {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10 nfcv 2256 . . . . 5 𝑦𝑥
11 dfrnf.2 . . . . 5 𝑦𝐴
12 nfcv 2256 . . . . 5 𝑦𝑤
1310, 11, 12nfbr 3942 . . . 4 𝑦 𝑥𝐴𝑤
1413nfex 1599 . . 3 𝑦𝑥 𝑥𝐴𝑤
15 nfv 1491 . . 3 𝑤𝑥 𝑥𝐴𝑦
16 breq2 3901 . . . 4 (𝑤 = 𝑦 → (𝑥𝐴𝑤𝑥𝐴𝑦))
1716exbidv 1779 . . 3 (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
1814, 15, 17cbvab 2238 . 2 {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
191, 9, 183eqtri 2140 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wex 1451  {cab 2101  wnfc 2243   class class class wbr 3897  ran crn 4508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  rnopab  4754
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